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Bunuel
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If 4^16 + 1 is divisible by positive integer x, then which of the foll 14 Apr 2020, 10:00
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If \(4^{16} + 1\) is divisible by positive integer x, then which of the following must also be divisible by x?


A. \(4^{96} + 1\)

B. \(4^{48} + 1\)

C. \(4^{32} + 1\)

D. \(4^{16} - 1\)

E. \(4^8 + 1\)

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If 4^16 + 1 is divisible by positive integer x, then which of the foll 14 Apr 2020, 11:52
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Bunuel
If \(4^{16} + 1\) is divisible by positive integer x, then which of the following must also be divisible by x?


A. \(4^{96} + 1\)

B. \(4^{48} + 1\)

C. \(4^{32} + 1\)

D. \(4^{16} - 1\)

E. \(4^8 + 1\)

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y^3 + 1 = (y+1)(y^2 - y +1)

If (y+1) is divisible by x - >( y^3 + 1) will also be divisible by x

4^{16*3} + 1 = 4^48 + 1 will also be divisible by x

IMO B
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If 4^16 + 1 is divisible by positive integer x, then which of the foll 14 Apr 2020, 20:54
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a^3 +b^3 =(a+b)(a^2-ab+b^2)

4^48+1= (4^16)^3 +1^3 = (4^16+1)× some integer

Therefore option B is also divisible by x

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If 4^16 + 1 is divisible by positive integer x, then which of the foll 14 Apr 2020, 23:14
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If one doesnt remember the formula for a^3+b^3 is there another approach?
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gurmukh
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If 4^16 + 1 is divisible by positive integer x, then which of the foll 15 Apr 2020, 00:11
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Kritisood
If one doesnt remember the formula for a^3+b^3 is there another approach?
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The point is not to know remember the formula because that anyway you can multiply and get the answer. The point is whether you can view the sum from that angle or not. If you remember the value of (a^3+b^3) then that helps you to develop the approach to solve the sum.

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If 4^16 + 1 is divisible by positive integer x, then which of the foll Updated on: 04 Jun 2020, 07:32
Originally posted by exc4libur on 03 Jun 2020, 09:33.
Last edited by exc4libur on 04 Jun 2020, 07:32, edited 1 time in total.
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Bunuel
If \(4^{16} + 1\) is divisible by positive integer x, then which of the following must also be divisible by x?

A. \(4^{96} + 1\)
B. \(4^{48} + 1\)
C. \(4^{32} + 1\)
D. \(4^{16} - 1\)
E. \(4^8 + 1\)
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(4^(16))^3+(1)^3=(a+b)(a^2+b^2-ab)
(4^(16)+1)((4^(16))^2+1-4^(16))
a+b is div by x

ans (B)
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If 4^16 + 1 is divisible by positive integer x, then which of the foll 03 Jun 2020, 09:56
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exc4libur
Bunuel
If \(4^{16} + 1\) is divisible by positive integer x, then which of the following must also be divisible by x?

A. \(4^{96} + 1\)
B. \(4^{48} + 1\)
C. \(4^{32} + 1\)
D. \(4^{16} - 1\)
E. \(4^8 + 1\)

\((4^{16} + 1)/x=(2^{32}+1)/x=p*x\)

\(4^{16} - 1=2^{32}-1=2^{32}-1^2=(2^{32}-1)(2^{32}+1)\)

\((2^{32}-1)(2^{32}+1)/x=p*x*(2^{32}-1)\)

Isn't (D) also correct?
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Check your calculation, there is mistake in it..

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If 4^16 + 1 is divisible by positive integer x, then which of the foll 22 Mar 2025, 13:15
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One approach if you don't remember the a^3 - b^3 formula :-

\(4^{16} + 1 = mx\) (some multiple of x)
\(4^{16} = mx - 1\)

Now, let's check the options:

For option B: \(4^{48} + 1 = 4^{16 \times 3} + 1 = (mx - 1)^3 + 1\)

Expanding \((mx - 1)^3\), we get:
\((mx - 1)^3 = m^3x^3 - 3m^2x^2 + 3mx - 1\)

So, \((mx - 1)^3 + 1 = m^3x^3 - 3m^2x^2 + 3mx - 1 + 1 = m^3x^3 - 3m^2x^2 + 3mx\)

This is clearly a multiple of x, so \(4^{48} + 1\) must also be divisible by x.

Therefore, the correct answer is B: \(4^{48} + 1\).
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